// ---------------------------------------------------------------------
//
// Copyright (C) 2017 - 2022 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
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#ifndef dealii_symmetric_tensor_templates_h
#define dealii_symmetric_tensor_templates_h


#include <deal.II/base/config.h>

#include <deal.II/base/exceptions.h>
#include <deal.II/base/symmetric_tensor.h>

#include <deal.II/differentiation/ad.h>

#include <deal.II/physics/transformations.h>

#include <array>
#include <limits>

DEAL_II_NAMESPACE_OPEN



template <typename Number>
std::array<Number, 1>
eigenvalues(const SymmetricTensor<2, 1, Number> &T)
{
  return {{T[0][0]}};
}



template <typename Number>
std::array<Number, 2>
eigenvalues(const SymmetricTensor<2, 2, Number> &T)
{
  // Make things work with AD types
  using std::sqrt;

  const Number upp_tri_sq = T[0][1] * T[0][1];
  if (upp_tri_sq == internal::NumberType<Number>::value(0.0))
    {
      // The tensor is diagonal
      std::array<Number, 2> eig_vals = {{T[0][0], T[1][1]}};

      // Sort from largest to smallest.
      std::sort(eig_vals.begin(), eig_vals.end(), std::greater<Number>());
      return eig_vals;
    }
  else
    {
      const Number tr_T    = trace(T);
      const Number det_T   = determinant(T);
      const Number descrim = tr_T * tr_T - 4.0 * det_T;
      Assert(
        descrim > internal::NumberType<Number>::value(0.0),
        ExcMessage(
          "The roots of the characteristic polynomial are complex valued."));
      const Number sqrt_desc = sqrt(descrim);

      const std::array<Number, 2> eig_vals = {
        {internal::NumberType<Number>::value(0.5 * (tr_T + sqrt_desc)),
         internal::NumberType<Number>::value(0.5 * (tr_T - sqrt_desc))}};
      Assert(eig_vals[0] >= eig_vals[1],
             ExcMessage("The eigenvalue ordering is incorrect."));
      return eig_vals;
    }
}



template <typename Number>
std::array<Number, 3>
eigenvalues(const SymmetricTensor<2, 3, Number> &T)
{
  // Make things work with AD types
  using std::acos;
  using std::cos;
  using std::sqrt;

  const Number upp_tri_sq =
    T[0][1] * T[0][1] + T[0][2] * T[0][2] + T[1][2] * T[1][2];
  if (upp_tri_sq == internal::NumberType<Number>::value(0.0))
    {
      // The tensor is diagonal
      std::array<Number, 3> eig_vals = {{T[0][0], T[1][1], T[2][2]}};

      // Sort from largest to smallest.
      std::sort(eig_vals.begin(), eig_vals.end(), std::greater<Number>());
      return eig_vals;
    }
  else
    {
      // Perform an affine change to T, and solve a different
      // characteristic equation that has a trigonometric solution.
      // Decompose T = p*B + q*I , and set q = tr(T)/3
      // and p = (tr((T - q.I)^{2})/6)^{1/2} . Then solve the equation
      // 0 = det(\lambda*I - B) = \lambda^{3} - 3*\lambda - det(B)
      // which has the solution
      // \lambda = 2*cos(1/3 * acos(det(B)/2) +2/3*pi*k ); k = 0,1,2
      // when substituting  \lambda = 2.cos(theta) and using trig identities.
      const Number tr_T = trace(T);
      const Number q    = tr_T / 3.0;
      const Number tmp1 = (T[0][0] - q) * (T[0][0] - q) +
                          (T[1][1] - q) * (T[1][1] - q) +
                          (T[2][2] - q) * (T[2][2] - q) + 2.0 * upp_tri_sq;
      const Number                        p = sqrt(tmp1 / 6.0);
      const SymmetricTensor<2, 3, Number> B =
        Number(1.0 / p) * (T - q * unit_symmetric_tensor<3, Number>());
      const Number tmp_2 = determinant(B) / 2.0;

      // The value of tmp_2 should be within [-1,1], however
      // floating point errors might place it slightly outside
      // this range. It is therefore necessary to correct for it.
      // Note: The three results in the conditional may lead to different
      //       number types when using Sacado numbers, so we cast them when
      //       necessary to a consistent result type.
      const Number phi =
        (tmp_2 <= -1.0 ?
           internal::NumberType<Number>::value(numbers::PI / 3.0) :
           (tmp_2 >= 1.0 ?
              internal::NumberType<Number>::value(0.0) :
              internal::NumberType<Number>::value(acos(tmp_2) / 3.0)));

      // Due to the trigonometric solution, the computed eigenvalues
      // should be predictably in the order eig1 >= eig2 >= eig3...
      std::array<Number, 3> eig_vals = {
        {static_cast<Number>(q + 2.0 * p * cos(phi)),
         static_cast<Number>(0.0),
         static_cast<Number>(q +
                             2.0 * p * cos(phi + (2.0 / 3.0 * numbers::PI)))}};
      // Use the identity tr(T) = eig1 + eig2 + eig3
      eig_vals[1] = tr_T - eig_vals[0] - eig_vals[2];

      // ... however, when equal roots exist then floating point
      // errors may make this no longer be the case.
      // Sort from largest to smallest.
      std::sort(eig_vals.begin(), eig_vals.end(), std::greater<Number>());

      return eig_vals;
    }
}



namespace internal
{
  namespace SymmetricTensorImplementation
  {
    template <int dim, typename Number>
    void
    tridiagonalize(const dealii::SymmetricTensor<2, dim, Number> &A,
                   dealii::Tensor<2, dim, Number> &               Q,
                   std::array<Number, dim> &                      d,
                   std::array<Number, dim - 1> &                  e)
    {
      // Make things work with AD types
      using std::sqrt;

      // Create some intermediate storage
      Number h, g, omega_inv, K, f;

      // Initialize the transformation matrix as the
      // identity tensor
      Q = static_cast<Tensor<2, dim, Number>>(
        dealii::unit_symmetric_tensor<dim, Number>());

      // Make the first row and column to be of the
      // desired form
      h = 0.0;
      for (int i = 1; i < dim; ++i)
        h += A[0][i] * A[0][i];

      g = 0.0;
      if (A[0][1] > 0.0)
        g = -sqrt(h);
      else
        g = sqrt(h);
      e[0] = g;

      std::array<Number, dim> u;
      for (int i = 1; i < dim; ++i)
        {
          u[i] = A[0][i];
          if (i == 1)
            u[i] -= g;
        }

      std::array<Number, dim> q;
      const Number            omega = h - g * A[0][1];
      if (omega > 0.0)
        {
          omega_inv = 1.0 / omega;
          K         = 0.0;
          for (int i = 1; i < dim; ++i)
            {
              f = 0.0;
              for (int j = 1; j < dim; ++j)
                f += A[i][j] * u[j];
              q[i] = omega_inv * f;
              K += u[i] * f;
            }
          K *= 0.5 * omega_inv * omega_inv;

          for (int i = 1; i < dim; ++i)
            q[i] = q[i] - K * u[i];

          d[0] = A[0][0];
          for (int i = 1; i < dim; ++i)
            d[i] = A[i][i] - 2.0 * q[i] * u[i];

          // Store inverse Householder transformation
          // in Q
          for (int j = 1; j < dim; ++j)
            {
              f = omega_inv * u[j];
              for (int i = 1; i < dim; ++i)
                Q[i][j] = Q[i][j] - f * u[i];
            }

          // For dim = 3: Calculate updated A[1][2] and
          // store it in e[1]
          for (int i = 1; i < dim - 1; ++i)
            e[i] = A[i][i + 1] - q[i] * u[i + 1] - u[i] * q[i + 1];
        }
      else
        {
          for (unsigned int i = 0; i < dim; ++i)
            d[i] = A[i][i];

          // For dim = 3:
          for (int i = 1; i < dim - 1; ++i)
            e[i] = A[i][i + 1];
        }
    }



    template <int dim, typename Number>
    std::array<std::pair<Number, Tensor<1, dim, Number>>, dim>
    ql_implicit_shifts(const dealii::SymmetricTensor<2, dim, Number> &A)
    {
      // Make things work with AD types
      using std::fabs;
      using std::sqrt;

      static_assert(
        numbers::NumberTraits<Number>::is_complex == false,
        "This implementation of the QL implicit shift algorithm does "
        "not support complex numbers");

      // Transform A to real tridiagonal form by the Householder method:
      // The orthogonal matrix effecting the transformation
      // this will ultimately store the eigenvectors
      dealii::Tensor<2, dim, Number> Q;
      // The diagonal elements of the tridiagonal matrix;
      // this will ultimately store the eigenvalues
      std::array<Number, dim> w;
      // The off-diagonal elements of the tridiagonal
      std::array<Number, dim - 1> ee;
      tridiagonalize<dim, Number>(A, Q, w, ee);

      // Number of iterations
      const unsigned int max_n_it = 30;

      // Transfer the off-diagonal entries to an auxiliary array
      // The third element is used only as temporary workspace
      std::array<Number, dim> e;
      for (unsigned int i = 0; i < dim - 1; ++i)
        e[i] = ee[i];

      // Create some intermediate storage
      Number g, r, p, f, b, s, c, t;

      // Loop over all off-diagonal elements
      for (int l = 0; l < dim - 1; ++l)
        {
          for (unsigned int it = 0; it <= max_n_it; ++it)
            {
              // Check for convergence and exit iteration loop
              // if the off-diagonal element e[l] is zero
              int m = l;
              for (; m <= dim - 2; ++m)
                {
                  g = fabs(w[m]) + fabs(w[m + 1]);
                  if (fabs(e[m]) + g == g)
                    break;
                }
              if (m == l)
                break;

              // Throw if no convergence is achieved within a
              // stipulated number of iterations
              if (it == max_n_it)
                {
                  AssertThrow(
                    false,
                    ExcMessage(
                      "No convergence in iterative QL eigenvector algorithm.")) return std::
                    array<std::pair<Number, Tensor<1, dim, Number>>, dim>();
                }

              // Calculate the shift..
              g = (w[l + 1] - w[l]) / (e[l] + e[l]);
              r = sqrt(g * g + 1.0);
              // .. and then compute g = d_m - k_s for the
              // plane rotation (Press2007a eq 11.4.22)
              if (g > 0.0)
                g = w[m] - w[l] + e[l] / (g + r);
              else
                g = w[m] - w[l] + e[l] / (g - r);

              // Perform plane rotation, as is done in the
              // standard QL algorithm, followed by Givens
              // rotations to recover the tridiagonal form
              s = c = 1.0;
              p     = 0.0;
              for (int i = m - 1; i >= l; i--)
                {
                  f = s * e[i];
                  b = c * e[i];

                  // Branch to recover from underflow
                  if (fabs(f) > fabs(g))
                    {
                      c        = g / f;
                      r        = sqrt(c * c + 1.0);
                      e[i + 1] = f * r;
                      c *= (s = 1.0 / r);
                    }
                  else
                    {
                      s        = f / g;
                      r        = sqrt(s * s + 1.0);
                      e[i + 1] = g * r;
                      s *= (c = 1.0 / r);
                    }

                  g        = w[i + 1] - p;
                  r        = (w[i] - g) * s + 2.0 * c * b;
                  p        = s * r;
                  w[i + 1] = g + p;
                  g        = c * r - b;

                  // Form the eigenvectors
                  for (unsigned int k = 0; k < dim; ++k)
                    {
                      t           = Q[k][i + 1];
                      Q[k][i + 1] = s * Q[k][i] + c * t;
                      Q[k][i]     = c * Q[k][i] - s * t;
                    }
                }
              w[l] -= p;
              e[l] = g;
              e[m] = 0.0;
            }
        }

      // Structure the data to be outputted
      std::array<std::pair<Number, Tensor<1, dim, Number>>, dim> eig_vals_vecs;
      for (unsigned int e = 0; e < dim; ++e)
        {
          eig_vals_vecs[e].first = w[e];

          // The column "e" of Q contains the non-normalized
          // eigenvector associated with the eigenvalue "e"
          for (unsigned int a = 0; a < dim; ++a)
            {
              eig_vals_vecs[e].second[a] = Q[a][e];
            }

          // Normalize
          Assert(eig_vals_vecs[e].second.norm() != 0.0, ExcDivideByZero());
          eig_vals_vecs[e].second /= eig_vals_vecs[e].second.norm();
        }
      return eig_vals_vecs;
    }



    template <int dim, typename Number>
    std::array<std::pair<Number, Tensor<1, dim, Number>>, dim>
    jacobi(dealii::SymmetricTensor<2, dim, Number> A)
    {
      static_assert(numbers::NumberTraits<Number>::is_complex == false,
                    "This implementation of the Jacobi algorithm does "
                    "not support complex numbers");

      // Make things work with AD types
      using std::fabs;
      using std::sqrt;

      // Sums of diagonal resp. off-diagonal elements
      Number sd, so;
      // sin(phi), cos(phi), tan(phi) and temporary storage
      Number s, c, t;
      // More temporary storage
      Number g, h, z, theta;
      // Threshold value
      Number thresh;

      // Initialize the transformation matrix as the
      // identity tensor
      dealii::Tensor<2, dim, Number> Q(
        dealii::unit_symmetric_tensor<dim, Number>());

      // The diagonal elements of the tridiagonal matrix;
      // this will ultimately store the eigenvalues
      std::array<Number, dim> w;
      for (unsigned int i = 0; i < dim; ++i)
        w[i] = A[i][i];

      // Calculate (tr(A))^{2}
      sd = trace(A);
      sd *= sd;

      // Number of iterations
      const unsigned int max_n_it = 50;
      for (unsigned int it = 0; it <= max_n_it; ++it)
        {
          // Test for convergence
          so = 0.0;
          for (unsigned int p = 0; p < dim; ++p)
            for (int q = p + 1; q < dim; ++q)
              so += fabs(A[p][q]);
          if (so == 0.0)
            break;

          // Throw if no convergence is achieved within a
          // stipulated number of iterations
          if (it == max_n_it)
            {
              AssertThrow(
                false,
                ExcMessage(
                  "No convergence in iterative Jacobi eigenvector algorithm.")) return std::
                array<std::pair<Number, Tensor<1, dim, Number>>, dim>();
            }

          // Compute threshold value which dictates whether or
          // not a Jacobi rotation is performed
          const unsigned int n_it_skip = 4;
          if (it < n_it_skip)
            thresh = 0.2 * so / (dim * dim);
          else
            thresh = 0.0;

          // Perform sweep
          for (unsigned int p = 0; p < dim; ++p)
            for (unsigned int q = p + 1; q < dim; ++q)
              {
                g = 100.0 * fabs(A[p][q]);

                // After a given number of iterations the
                // rotation is skipped if the off-diagonal
                // element is small
                if (it > n_it_skip && fabs(w[p]) + g == fabs(w[p]) &&
                    fabs(w[q]) + g == fabs(w[q]))
                  {
                    A[p][q] = 0.0;
                  }
                else if (fabs(A[p][q]) > thresh)
                  {
                    // Calculate Jacobi transformation
                    h = w[q] - w[p];

                    // Compute surrogate for angle theta resulting from
                    // angle transformation and subsequent smallest solution
                    // of quadratic equation
                    if (fabs(h) + g == fabs(h))
                      {
                        // Prevent overflow for large theta^2. This computation
                        // is the algebraic equivalent of t = 1/(2*theta).
                        t = A[p][q] / h;
                      }
                    else
                      {
                        theta = 0.5 * h / A[p][q];
                        if (theta < 0.0)
                          t = -1.0 / (sqrt(1.0 + theta * theta) - theta);
                        else
                          t = 1.0 / (sqrt(1.0 + theta * theta) + theta);
                      }

                    // Compute trigonometric functions for rotation
                    // in such a way as to prevent overflow for
                    // large theta.
                    c = 1.0 / sqrt(1.0 + t * t);
                    s = t * c;
                    z = t * A[p][q];

                    // Apply Jacobi transformation...
                    A[p][q] = 0.0;
                    w[p] -= z;
                    w[q] += z;
                    // ... by executing the various rotations in sequence
                    for (unsigned int r = 0; r < p; ++r)
                      {
                        t       = A[r][p];
                        A[r][p] = c * t - s * A[r][q];
                        A[r][q] = s * t + c * A[r][q];
                      }
                    for (unsigned int r = p + 1; r < q; ++r)
                      {
                        t       = A[p][r];
                        A[p][r] = c * t - s * A[r][q];
                        A[r][q] = s * t + c * A[r][q];
                      }
                    for (unsigned int r = q + 1; r < dim; ++r)
                      {
                        t       = A[p][r];
                        A[p][r] = c * t - s * A[q][r];
                        A[q][r] = s * t + c * A[q][r];
                      }

                    // Update the eigenvectors
                    for (unsigned int r = 0; r < dim; ++r)
                      {
                        t       = Q[r][p];
                        Q[r][p] = c * t - s * Q[r][q];
                        Q[r][q] = s * t + c * Q[r][q];
                      }
                  }
              }
        }

      // Structure the data to be outputted
      std::array<std::pair<Number, Tensor<1, dim, Number>>, dim> eig_vals_vecs;
      for (unsigned int e = 0; e < dim; ++e)
        {
          eig_vals_vecs[e].first = w[e];

          // The column "e" of Q contains the non-normalized
          // eigenvector associated with the eigenvalue "e"
          for (unsigned int a = 0; a < dim; ++a)
            {
              eig_vals_vecs[e].second[a] = Q[a][e];
            }

          // Normalize
          Assert(eig_vals_vecs[e].second.norm() != 0.0, ExcDivideByZero());
          eig_vals_vecs[e].second /= eig_vals_vecs[e].second.norm();
        }
      return eig_vals_vecs;
    }



    template <typename Number>
    std::array<std::pair<Number, Tensor<1, 2, Number>>, 2>
    hybrid(const dealii::SymmetricTensor<2, 2, Number> &A)
    {
      static_assert(numbers::NumberTraits<Number>::is_complex == false,
                    "This implementation of the 2d Hybrid algorithm does "
                    "not support complex numbers");

      // Make things work with AD types
      using std::fabs;

      const unsigned int dim = 2;

      // Calculate eigenvalues
      const std::array<Number, dim> w = eigenvalues(A);

      std::array<std::pair<Number, Tensor<1, dim, Number>>, dim> eig_vals_vecs;

      Number t, u; // Intermediate storage
      t = fabs(w[0]);
      for (unsigned int i = 1; i < dim; ++i)
        {
          u = fabs(w[i]);
          if (u > t)
            t = u;
        }

      if (t < 1.0)
        u = t;
      else
        u = t * t;

      // Estimated maximum roundoff error
      const Number error =
        256.0 * std::numeric_limits<double>::epsilon() * u * u;

      // Store eigenvalues
      eig_vals_vecs[0].first = w[0];
      eig_vals_vecs[1].first = w[1];

      // Compute eigenvectors
      // http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/
      // https://math.stackexchange.com/a/1548616
      if (A[1][0] != 0.0)
        {
          // First eigenvector
          eig_vals_vecs[0].second[0] = w[0] - A[1][1];
          eig_vals_vecs[0].second[1] = A[1][0];

          // Second eigenvector
          eig_vals_vecs[1].second[0] = w[1] - A[1][1];
          eig_vals_vecs[1].second[1] = A[1][0];
        }
      else
        {
          // First eigenvector
          eig_vals_vecs[0].second[0] = w[0];
          eig_vals_vecs[0].second[1] = 0.0;

          // Second eigenvector
          eig_vals_vecs[1].second[0] = 0.0;
          eig_vals_vecs[1].second[1] = w[1];
        }
      // Normalize
      eig_vals_vecs[0].second /= eig_vals_vecs[0].second.norm();
      eig_vals_vecs[1].second /= eig_vals_vecs[1].second.norm();

      // If vectors are nearly linearly dependent, or if there might have
      // been large cancellations in the calculation of A[i][i] - w[0], fall
      // back to QL algorithm
      if (eig_vals_vecs[0].second * eig_vals_vecs[1].second > error)
        {
          return ql_implicit_shifts(A);
        }

      return eig_vals_vecs;
    }



    template <typename Number>
    std::array<std::pair<Number, Tensor<1, 3, Number>>, 3>
    hybrid(const dealii::SymmetricTensor<2, 3, Number> &A)
    {
      static_assert(numbers::NumberTraits<Number>::is_complex == false,
                    "This implementation of the 3d Hybrid algorithm does "
                    "not support complex numbers");

      // Make things work with AD types
      using std::fabs;
      using std::sqrt;

      const unsigned int dim = 3;
      Number norm; // Squared norm or inverse norm of current eigenvector
      Number t, u; // Intermediate storage

      // Calculate eigenvalues
      const std::array<Number, dim> w = eigenvalues(A);

      t = fabs(w[0]);
      for (unsigned int i = 1; i < dim; ++i)
        {
          u = fabs(w[i]);
          if (u > t)
            t = u;
        }

      if (t < 1.0)
        u = t;
      else
        u = t * t;

      // Estimated maximum roundoff error
      const Number error =
        256.0 * std::numeric_limits<double>::epsilon() * u * u;

      // Initialize the transformation matrix as the
      // identity tensor
      dealii::Tensor<2, dim, Number> Q;
      Q[0][1] = A[0][1] * A[1][2] - A[0][2] * A[1][1];
      Q[1][1] = A[0][2] * A[0][1] - A[1][2] * A[0][0];
      Q[2][1] = A[0][1] * A[0][1];

      // Calculate first eigenvector by the formula
      //   v[0] = (A - w[0]).e1 x (A - w[0]).e2
      Q[0][0] = Q[0][1] + A[0][2] * w[0];
      Q[1][0] = Q[1][1] + A[1][2] * w[0];
      Q[2][0] = (A[0][0] - w[0]) * (A[1][1] - w[0]) - Q[2][1];
      norm    = Q[0][0] * Q[0][0] + Q[1][0] * Q[1][0] + Q[2][0] * Q[2][0];

      // If vectors are nearly linearly dependent, or if there might have
      // been large cancellations in the calculation of A[i][i] - w[0], fall
      // back to QL algorithm
      // Note that this simultaneously ensures that multiple eigenvalues do
      // not cause problems: If w[0] = w[1], then A - w[0] * I has rank 1,
      // i.e. all columns of A - w[0] * I are linearly dependent.
      if (norm <= error)
        {
          return ql_implicit_shifts(A);
        }
      else // This is the standard branch
        {
          norm = sqrt(1.0 / norm);
          for (unsigned j = 0; j < dim; ++j)
            Q[j][0] = Q[j][0] * norm;
        }

      // Calculate second eigenvector by the formula
      //   v[1] = (A - w[1]).e1 x (A - w[1]).e2
      Q[0][1] = Q[0][1] + A[0][2] * w[1];
      Q[1][1] = Q[1][1] + A[1][2] * w[1];
      Q[2][1] = (A[0][0] - w[1]) * (A[1][1] - w[1]) - Q[2][1];
      norm    = Q[0][1] * Q[0][1] + Q[1][1] * Q[1][1] + Q[2][1] * Q[2][1];
      if (norm <= error)
        {
          return ql_implicit_shifts(A);
        }
      else
        {
          norm = sqrt(1.0 / norm);
          for (unsigned int j = 0; j < dim; ++j)
            Q[j][1] = Q[j][1] * norm;
        }

      // Calculate third eigenvector according to
      //   v[2] = v[0] x v[1]
      Q[0][2] = Q[1][0] * Q[2][1] - Q[2][0] * Q[1][1];
      Q[1][2] = Q[2][0] * Q[0][1] - Q[0][0] * Q[2][1];
      Q[2][2] = Q[0][0] * Q[1][1] - Q[1][0] * Q[0][1];

      // Structure the data to be outputted
      std::array<std::pair<Number, Tensor<1, dim, Number>>, dim> eig_vals_vecs;
      for (unsigned int e = 0; e < dim; ++e)
        {
          eig_vals_vecs[e].first = w[e];

          // The column "e" of Q contains the non-normalized
          // eigenvector associated with the eigenvalue "e"
          for (unsigned int a = 0; a < dim; ++a)
            {
              eig_vals_vecs[e].second[a] = Q[a][e];
            }

          // Normalize
          Assert(eig_vals_vecs[e].second.norm() != 0.0, ExcDivideByZero());
          eig_vals_vecs[e].second /= eig_vals_vecs[e].second.norm();
        }
      return eig_vals_vecs;
    }


    template <typename Number>
    Tensor<2, 1, Number>
    dediagonalize_tensor(const dealii::SymmetricTensor<2, 1, Number> &T,
                         const double /*rotation_angle*/,
                         const unsigned int /*axis*/ = 0)
    {
      AssertThrow(false, ExcNotImplemented());
      return Tensor<2, 1, Number>({{T[0][0]}});
    }


    template <typename Number>
    Tensor<2, 2, Number>
    dediagonalize_tensor(const dealii::SymmetricTensor<2, 2, Number> &T,
                         const double rotation_angle,
                         const unsigned int /*axis*/ = 0)
    {
      const Tensor<2, 2> R =
        dealii::Physics::Transformations::Rotations::rotation_matrix_2d(
          rotation_angle);
      return R * T;
    }


    template <typename Number>
    Tensor<2, 3, Number>
    dediagonalize_tensor(const dealii::SymmetricTensor<2, 3, Number> &T,
                         const double       rotation_angle,
                         const unsigned int axis = 0)
    {
      AssertIndexRange(axis, 3);

      Tensor<2, 3> R;
      switch (axis)
        {
          case (0):
            R = dealii::Physics::Transformations::Rotations::rotation_matrix_3d(
              Tensor<1, 3>({1., 0., 0.}), rotation_angle);
            break;
          case (1):
            R = dealii::Physics::Transformations::Rotations::rotation_matrix_3d(
              Tensor<1, 3>({0., 1., 0.}), rotation_angle);
            break;
          case (2):
            R = dealii::Physics::Transformations::Rotations::rotation_matrix_3d(
              Tensor<1, 3>({0., 0., 1.}), rotation_angle);
            break;
          default:
            AssertThrow(false, ExcNotImplemented());
            break;
        }
      return R * T;
    }


    template <typename Number>
    std::array<std::pair<Number, Tensor<1, 1, Number>>, 1>
    perform_eigenvector_decomposition(
      const SymmetricTensor<2, 1, Number> &T,
      const SymmetricTensorEigenvectorMethod /*method*/)
    {
      return {{std::make_pair(T[0][0], Tensor<1, 1, Number>({1.0}))}};
    }


    template <int dim, typename Number>
    std::array<std::pair<Number, Tensor<1, dim, Number>>, dim>
    perform_eigenvector_decomposition(
      const SymmetricTensor<2, dim, Number> &T,
      const SymmetricTensorEigenvectorMethod method)
    {
      switch (method)
        {
          case SymmetricTensorEigenvectorMethod::hybrid:
            return internal::SymmetricTensorImplementation::hybrid(T);
            break;
          case SymmetricTensorEigenvectorMethod::ql_implicit_shifts:
            return internal::SymmetricTensorImplementation::ql_implicit_shifts(
              T);
            break;
          case SymmetricTensorEigenvectorMethod::jacobi:
            return internal::SymmetricTensorImplementation::jacobi(T);
            break;
          default:
            break;
        }

      AssertThrow(false, ExcNotImplemented());
      return std::array<std::pair<Number, Tensor<1, dim, Number>>, dim>();
    }


  } // namespace SymmetricTensorImplementation
} // namespace internal



template <int dim, typename Number>
std::array<std::pair<Number, Tensor<1, dim, Number>>,
           std::integral_constant<int, dim>::value>
eigenvectors(const SymmetricTensor<2, dim, Number> &T,
             const SymmetricTensorEigenvectorMethod method)
{
  // Not much to do when there's only a single entry
  if (dim == 1)
    return internal::SymmetricTensorImplementation::
      perform_eigenvector_decomposition(T, method);

  std::array<std::pair<Number, Tensor<1, dim, Number>>, dim> eig_vals_vecs;

  if (Differentiation::AD::is_ad_number<Number>::value && dim > 1)
    {
      // If the tensor is diagonal, then we have a bit on an issue when using
      // auto-differentiable numbers. The reason for this is that all of the
      // algorithms have shortcuts by which to return result in this case.
      // This artificially decouples the eigenvalues/vectors, which upon
      // differentiation leads to the wrong result (each is, incorrectly,
      // insensitive with respect to the other). To work around this manipulate
      // tensor @p T in an objective manner: through an infinitesimal rotation we
      // make it non-diagonal (although we introduce some numerical error).
      bool is_diagonal = true;
      for (unsigned int i = 0; i < dim; ++i)
        for (unsigned int j = i + 1; j < dim; ++j)
          if (T[i][j] != 0.0)
            {
              is_diagonal = false;
              break;
            }

      // If our tensor is not diagonal, then just carry on as per usual.
      if (!is_diagonal)
        eig_vals_vecs = internal::SymmetricTensorImplementation::
          perform_eigenvector_decomposition(T, method);
      else
        {
          Assert(
            method != SymmetricTensorEigenvectorMethod::hybrid,
            ExcMessage(
              "The hybrid method cannot be used with auto-differentiable numbers "
              "when the tensor upon which an eigen-decomposition is being performed "
              "is diagonal. This is because the hybrid method immediately assumes "
              "the values of the eigenvectors (since the characteristic polynomial) "
              "is not solved, and therefore the sensitivity of the eigenvalues with "
              "respect to one another is not resolved."));

          // These parameters are heuristicaly chosen through "rigorous"
          // eye-ball analysis of the errors of tests based on
          // ad-common-tests/symmetric_tensor_functions_03.h. This checks the
          // first and second derivatives of the representation of a Neo-Hookean
          // material described by an Ogden-type model (i.e. using an
          // eigen-decomposition of the right Cauchy-Green tensor). Using this
          // comparison between the well-understood result expected from the
          // Neo-Hookean model and its Ogden equivalent, these parameters are a
          // first approximation to those required to collectively minimize
          // error in the energy values, first and second derivatives. What's
          // apparent is that all AD numbers and eigen-decomposition algorithms
          // are not made equal!
          double sf = 1.0;
          if (Differentiation::AD::is_taped_ad_number<Number>::value)
            {
              // ADOL-C taped
              if (method ==
                  SymmetricTensorEigenvectorMethod::ql_implicit_shifts)
                sf = 2e11;
              else if (method == SymmetricTensorEigenvectorMethod::jacobi)
                sf = (dim == 2 ? 1e6 : 1e9);
              else
                AssertThrow(false, ExcNotImplemented());
            }
          else if (Differentiation::AD::is_sacado_rad_number<Number>::value)
            {
              // Sacado::Rad
              if (method ==
                  SymmetricTensorEigenvectorMethod::ql_implicit_shifts)
                sf = (dim == 2 ? 1e8 : 1e9);
              else if (method == SymmetricTensorEigenvectorMethod::jacobi)
                sf = (dim == 2 ? 1e8 : 1e9);
              else
                AssertThrow(false, ExcNotImplemented());
            }
          else
            {
              // Everything else
              Assert(Differentiation::AD::is_tapeless_ad_number<Number>::value,
                     ExcInternalError());
              Assert(
                Differentiation::AD::is_sacado_dfad_number<Number>::value ||
                  Differentiation::AD::is_adolc_tapeless_number<Number>::value,
                ExcInternalError());

              if (method ==
                  SymmetricTensorEigenvectorMethod::ql_implicit_shifts)
                sf = (dim == 2 ? 1e7 : 2.5e7);
              else if (method == SymmetricTensorEigenvectorMethod::jacobi)
                sf = (dim == 2 ? 1e2 : 1e7);
              else
                AssertThrow(false, ExcNotImplemented());
            }

          using scalar_type =
            typename Differentiation::AD::ADNumberTraits<Number>::scalar_type;
          const double delta = sf * std::numeric_limits<scalar_type>::epsilon();
          const double rotation_angle = delta * numbers::PI / 180.0;

          if (dim == 2)
            {
              const Tensor<2, dim, Number> T_prime_ns =
                internal::SymmetricTensorImplementation::dediagonalize_tensor(
                  T, rotation_angle);

              // We can't symmetrize the tensor, otherwise the sensitivities
              // cancel out. So we take the upper triangle as an approximation
              // instead.
              // TODO[JPP]: Perform the eigen-decomposition on the non-symmetric
              //            T_prime_ns. This is, however, nontrivial to
              //            implement in this context. See:
              //            http://www.alglib.net/eigen/nonsymmetric/nonsymmetricevd.php
              //            https://groups.google.com/forum/#!topic/stan-users/QJe1TNioiyg
              SymmetricTensor<2, dim, Number> T_prime;
              for (unsigned int i = 0; i < dim; ++i)
                for (unsigned int j = i; j < dim; ++j)
                  T_prime[i][j] = T_prime_ns[i][j];

              eig_vals_vecs = internal::SymmetricTensorImplementation::
                perform_eigenvector_decomposition(T_prime, method);
            }
          else
            {
              Assert(dim == 3, ExcDimensionMismatch(dim, 3));

              SymmetricTensor<2, dim, Number> T_prime;
              Tensor<2, dim, Number>          T_prime_ns;
              for (unsigned int i = 0; i < dim; ++i)
                {
                  // This is a little bit hacky, so here's a brief explanation
                  // as to what the principal of this operation is: What we're
                  // trying to do here is perturb our tensor such that the
                  // sensitivity of the eigenvectors with respect to each other
                  // can be established. So, one at a time, we compute the
                  // perturbation of the input tensor such that the maximal
                  // number of off-diagonal entries are non-zero for any given
                  // "i". This means that we rotation not about the "ith" axis,
                  // but rather some offset of it. Note: This does NOT lead to
                  // an exact value or derivative of the eigendata being
                  // computed, so one should be aware that for this case (where
                  // the eigenvalues are equal), the linearization of any
                  // resulting quantities is only approximate.
                  const unsigned int axis = (i + 2) % 3;
                  T_prime_ns = internal::SymmetricTensorImplementation::
                    dediagonalize_tensor(T, rotation_angle, axis);

                  // We can't symmetrize the tensor, otherwise the sensitivities
                  // cancel out. So we take the upper triangle as an
                  // approximation instead.
                  // TODO[JPP]: Keep the full row and perform the
                  //            eigen-decomposition on the
                  //            non-symmetric T_prime_ns. See the related
                  //            comment above in the 2d case.
                  for (unsigned int j = i; j < dim; ++j)
                    T_prime[i][j] = T_prime_ns[i][j];
                }
              eig_vals_vecs = internal::SymmetricTensorImplementation::
                perform_eigenvector_decomposition(T_prime, method);
            }
        }
    }
  else
    eig_vals_vecs = internal::SymmetricTensorImplementation::
      perform_eigenvector_decomposition(T, method);

  // Sort in descending order before output.
  std::sort(
    eig_vals_vecs.begin(),
    eig_vals_vecs.end(),
    internal::SymmetricTensorImplementation::SortEigenValuesVectors<dim,
                                                                    Number>());
  return eig_vals_vecs;
}



#ifdef DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING
namespace internal
{
  namespace SymmetricTensorImplementation
  {
    template <>
    struct Inverse<4, 3, adouble>;
  } // namespace SymmetricTensorImplementation
} // namespace internal

template <>
std::array<adouble, 1>
eigenvalues(const SymmetricTensor<2, 1, adouble> & /*T*/);

template <>
std::array<adouble, 2>
eigenvalues(const SymmetricTensor<2, 2, adouble> & /*T*/);

template <>
std::array<adouble, 3>
eigenvalues(const SymmetricTensor<2, 3, adouble> & /*T*/);

template <>
std::array<std::pair<adouble, Tensor<1, 1, adouble>>, 1>
eigenvectors(const SymmetricTensor<2, 1, adouble> & /*T*/,
             const SymmetricTensorEigenvectorMethod /*method*/);

template <>
std::array<std::pair<adouble, Tensor<1, 2, adouble>>, 2>
eigenvectors(const SymmetricTensor<2, 2, adouble> & /*T*/,
             const SymmetricTensorEigenvectorMethod /*method*/);

template <>
std::array<std::pair<adouble, Tensor<1, 3, adouble>>, 3>
eigenvectors(const SymmetricTensor<2, 3, adouble> & /*T*/,
             const SymmetricTensorEigenvectorMethod /*method*/);
#endif

DEAL_II_NAMESPACE_CLOSE

#endif
